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The motion of a layer of yield-stress material on an oscillating plate

Published online by Cambridge University Press:  22 March 2023

Edward M. Hinton Open the ORCID record for Edward M. Hinton [Opens in a new window] ,
Jesse F. Collis Open the ORCID record for Jesse F. Collis [Opens in a new window]  and
John E. Sader Open the ORCID record for John E. Sader [Opens in a new window]
Edward M. Hinton
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010 Australia
Jesse F. Collis
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010 Australia
John E. Sader*
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125 USA
*
†Email address for correspondence: jsader@caltech.edu
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Abstract

The motion of a finite layer of Bingham material on a solid plate that executes in-plane oscillations was reported previously by Balmforth et al. (J. Non-Newtonian Fluid Mech., vol. 158, issue 1–3, 2009, pp. 46–53). There, it was suggested that multiple yielded regions may arise within the material; this contrasts to start-up flow of the same material for which only one yielded region is generated. Here, we explore quantitatively the fluid physics of this oscillatory flow problem through analytical approximations and further numerical computation. Four new key topological properties concerning the generation and termination of the yielded regions are reported. It is shown that the existence of multiple yielded regions is equivalent to the layer never becoming entirely rigid during the periodic motion. For small inertia, the flow is approximately time-reversible with only a single yielded region generated at the plate. For large inertia, shear stress in the material decays rapidly as a function of distance from the plate. A thin zone of yielded material detaches periodically from the plate, and subsequently terminates within the layer. At high oscillation frequency, there can be any number $N$ of distinct rigid regions, satisfying $N= \lfloor 1- {\rm \pi}^{-1} \log B \rfloor$ where $B$ is the Bingham number. It is also shown that for $B>0.5370$, there are at most one yielded region and one rigid region throughout the motion. These theoretical results can be used as a basis for oscillatory rheometry, allowing for measurement of the yield stress using existing apparatus.

JFM classification

Non-Newtonian Flows: Plastic materials Non-Newtonian Flows: Rheology
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 959 , 25 March 2023 , A32
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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